Problem: A group of adults and kids went to see a movie. Tickets cost $$7.50$ each for adults and $$4.00$ each for kids, and the group paid $$70.00$ in total. There were $6$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Solution: Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${7.5x+4y = 70}$ ${x = y-6}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-6}$ for $x$ in the first equation. ${7.5}{(y-6)}{+ 4y = 70}$ Simplify and solve for $y$ $ 7.5y-45 + 4y = 70 $ $ 11.5y-45 = 70 $ $ 11.5y = 115 $ $ y = \dfrac{115}{11.5} $ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into ${x = y-6}$ to find $x$ ${x = }{(10)}{ - 6}$ ${x = 4}$ You can also plug ${y = 10}$ into ${7.5x+4y = 70}$ and get the same answer for $x$ ${7.5x + 4}{(10)}{= 70}$ ${x = 4}$ There were $4$ adults and $10$ kids.